Kaprekar's transformations. Part I - Theoretical discussion

The paper is devoted to discussion of the minimal cycles of the so called Kaprekar's transformations and some of its generalizations. The considered transformations are the self-maps of the sets of natural numbers possessing n digits in their decimal expansions. In the paper there are introduced several new characteristics of such maps, among others, the ones connected with the Sharkovsky's theorem and with the Erdo?s-Szekeres theorem concerning the monotonic subsequences. Because of the size the study is divided into two parts. Part I includes the considerations of strictly theoretical nature resulting from the definition of Kaprekar's transformations. We find here all the minimal orbits of Kaprekar's transformations T_n, for n = 3,..., 7. Moreover, we define many different generalizations of the Kaprekar's transformations and we discuss their minimal orbits for the selected cases. In Part II (ibidem), which is a continuation of the current paper, the theoretical discussion will be supported by the numerical observations. For example, we notice there that each fixed point, familiar to us, of any Kaprekar's transformation generates an infinite sequence of fixed points of the other Kaprekar's transformations. The observed facts concern also several generalizations of the Kaprekar's transformations defined in Part I.